O objetivo desta dissertaÃÃo à apresentar um resultado similar ao Teorema de Bernstein sobre hipersuperfÃcies mÃnimas no espaÃo euclidiano, isto Ã, mostrar que tal resultado se generaliza para hipersuperfÃcies de Sn+1 com curvatura mÃdia constante, cuja aplicaÃÃo de Gauss estÃcontida em um hemis- fÃrio fechado de Sn+1 (Teorema 3.1). PorÃm, no caso em que a hipersuperfÃcie à mÃnima, utilizaremos na demonstraÃÃo deste teorema, um resultado sobre caracterizaÃÃo das hiperesferas de Sn+1 entre todas hipersuperfÃcies de Sn+1 em termos de suas imagens de Gauss (Teorema 2.1). / The objective of this dissertation is to show a similar result of Bernstein theorem about minimal hypersurfaces in Euclidian space, that is, to show that that result is generalized to hypersurfaces of Sn+1 with constant mean curvature, whose Gauss image is contained in a closed hemisphere of Sn+1(Theorem 3.1). However, in the case where the hypersurface is minimal, we will use in the proof of this theorem a result about the characterization of the hyperspheres of Sn+1 among all complete hypersurfaces in Sn+1 in terms of their Gauss images (Theorem 2.1)

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho estudaremos o Ãndice de hipersuperfÃcies mÃnimas e de curvatura mÃdia constante imersas na esfera Euclidiana Sn+1. Mais precisamente, definiremos o operador de Jacobi de hipersuperfÃcies mÃnimas e de curvatura mÃdia constante usando as fÃrmulas de variaÃÃo de Ãrea, e em seguida estabeleceremos estimativas por baixo para o Ãndice de hipersuperfÃcies mÃnimas imersas em Sn+1 . AlÃm disso, caracterizaremos os toros de Clifford mÃnimos como as hipersuperfÃcies compactas, orientÃveis e mÃnimas em Sn+1 tais que a = -2n, onde a à o primeiro autovalor do operador de Jacobi. Mostraremos que as esferas totalmente umbÃlicas Sn (r) em Sn+1, com 0 < r < 1, sÃo as hipersuperfÃcies fracamente estÃveis em Sn+1. Por Ãltimo, estabeleceremos estimativas por baixo para o Ãndice fraco de hipersuperfÃcies de curvatura mÃdia constante em Sn+1 e caracterizaremos os toros de Clifford Sk (r) x Sn-k (1 - r2) de curvatura mÃdia constante como as hipersuperfÃcies de curvatura mÃdia constante tais que o Ãndice fraco à igual a n + 2, onde (k/n + 2 ) &#8804; r &#8804; (k + 2/n + 2) Â. / The aim of this work is to study the index either of compact minimal or constant mean curvature hypersurfaces immersed into the Euclidean unit sphere Sn+1. The main ingredient to do that is the Jacobi operator which appears on the second formula of variation of area. On the minimal case we shall present low estimative for the index and we shall show that the minimal Clifford tori are the unique minimal hypersurfaces over which a = -2n , where a stands for the first eigenvalue of the Jacobi operator. Moreover, it is easy to see that totally umbilical sphere Sn (r) em Sn+1 , with 0 < r < 1, are weakly stable. Finally we shall show that the index is bigger that or equal to n+2 for compact constant mean curvature hypersurfaces of Sn+1 provides they have constant scalar curvature. Moreover , Clifford tori Sk (r) x Sn-k (1 - r2) attain such index provided (k/n + 2 ) &#8804; r &#8804; (k + 2/n + 2) Â.

O presente trabalho apresenta resultados objetivando classificar folheaÃÃes de codimensÃo 1 em variedades Riemannianas cujas folhas tem curvatura mÃdia constante. O principal resultado à o teorema de Barbosa-Kenmotsu-Oshikiri([3]), Teorema: Seja M uma variedade Riemanniana compacta com curvatura de Ricci nÃo negativa e F um folheaÃÃo de codimensÃo 1 e classe C3 de M, transversalmente orientÃvel, cujas folhas tem curvatura mÃdia constante. EntÃo, qualquer folha de F à uma subvariedade totalmente geodÃsica de M. AlÃm disso, M à localmente um produto Riemanniano de uma folha de F e uma curva normal e a curvatura de Ricci na direÃÃo normal Ãs folhas à zero. O resultado anterior nÃo pode ser estendido para o caso onde M à nÃo compacta. Uma folheaÃÃo contra-exemplo pode ser construÃda a partir de uma funÃÃo f que nÃo satisfaz a conjectura de Bernstein. No final, sÃo apresentados resultados recentes sobre os problemas abordados e uma prova da desigualdade de Heinz-Chern / In this paper, we work showing results aiming classify foliations of codimension-one in Riemannian manifolds whose leaves have constant mean curvature. The main result is the theorem by Barbosa-Kenmotsu-Oshikiri([3]). Theorem: LetM be a compact Riemannian manifold with nonnegative Ricci curvature e F, a codimensiononeC3-foliation of M whose leaves have constant mean curvature. The any leaf of F is totally geodesic submanifold of M. Futhermore M is locally a Riemannian product of a leaf of F and a normal curve,and the Ricci curvature in the direction normal to the leaves is zero. The previous result can not be extended for the case where M is not compact. A foliation counterexample can be built from a function f that does not satisfy the Bernsteinâs conjecture. At the end, they are present recent results about the boarded problems and a proof of the Heinz-Chern inequality.

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In this work we study two problems about parabolic partial differential equations on Riemannian manifolds with conical singularities. The first problem we are concerned with is the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. By introducing so called weighted Hölder and Sobolev spaces with discrete asymptotics, we provide a complete existence and regularity theory for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. The second problem we study is the short time existence problem for the generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds, when the initial Lagrangian submanifold has isolated conical singularities that are modelled on stable special Lagrangian cones. First we use Lagrangian neighbourhood theorems for Lagrangian submanifolds with conical singularities to integrate the generalized Lagrangian mean curvature flow to a nonlinear parabolic equation of functions, and then, using the existence and regularity theory for the heat equation, we prove short time existence of the generalized Lagrangian mean curvature flow with isolated conical singularities by letting the conical singularities move around in the ambient space and the model cones to rotate by unitary transformations.

The purpose of this thesis is to further develop numerical methods for solving surface problems by utilizing tangential calculus and the trace finite element method. Direct computation on the surface is possible by the use of tangential calculus, in contrast to the classical approach of mapping 2D parametric surfaces to 3D surfaces by means of differential geometry operators. Using tangential calculus, the problem formulation is only dependent on the position and normal vectors of the 3D surface. Tangential calculus thus enables a clean, simple and inexpensive formulation and implementation of finite element methods for surface problems. Meshing techniques are greatly simplified from the end-user perspective by utilizing an unfitted finite element method called the Trace Finite Element Method, in which the basic idea is to embed the surface in a higher dimensional mesh and use the shape functions of this background mesh for the discretization of the partial differential equation. This method makes it possible to model surfaces implicitly and solve surface problems without the need for expensive meshing/re-meshing techniques especially for moving surfaces or surfaces embedded in 3D solids, so called embedded interface problems. Using these two approaches, numerical methods for solving three surface problems are proposed: 1) minimal surface problems, in which the form that minimizes the mean curvature was computed by iterative update of a level-set function discretized using TraceFEM and driven by advection, for which the velocity field was given by the mean curvature flow, 2) elastic membrane problems discretized using linear and higher order TraceFEM, which makes it straightforward to embed complex geometries of membrane models into an elastic bulk for reinforcement and 3) stabilized, accurate vertex normal and mean curvature estimation with local refinement on triangulated surfaces. In this thesis the basics of the two main approaches are presented, some aspects such as stabilization and surface reconstruction are further developed, evaluated and numerically analyzed, details on implementations are provided and the current state of work is presented.

Neste trabalho, estabelecemos uma equação que relaciona a curvatura de Ricci de uma variedade riemanniana M e as segundas formas fundamentais de duas folheações ortogonais de dimensões complementares, F e F, definidas em M. Usando essa equação, encontramos uma estimativa da curvatura média da folheação F e uma condição necessária e suficiente para que tal folheação seja totalmente geodésica. Mostramos também uma condição suficiente para que M seja localmente um produto riemanniano das folhas de F e F, se uma das folheações for totalmente umbílica. Por fim, provamos ainda uma fórmula integral válida para tais folheações. / In this work, we and an equation that relates the Ricci curvature of a riemannian manifold M and the second fundamental forms of two orthogonal foliations of complementary dimensions, F and F, defined on M. Using this equation, we and an estimate of the mean curvature of the foliation F and a necessary and suficient condition for the foliation F to be totally geodesic. We also show a suficient condition for the manifold M to be locally a riemannian product of the leaves of F and F, if one of the foliations is totally umbilical. Finally, we also prove an integral formula for such foliations.

In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.

In this thesis, we study the overfitting problem in supervised learning of classifiers from a geometric perspective. As with many inverse problems, learning a classification function from a given set of example-label pairs is an ill-posed problem, i.e., there exist infinitely many classification functions that can correctly predict the class labels for all training examples. Among them, according to Occam's razor, simpler functions are favored since they are less overfitted to training examples and are therefore expected to perform better on unseen examples. The standard technique to enforce Occam's razor is to introduce a regularization scheme, which penalizes some type of complexity of the learned classification function. Some widely used regularization techniques are functional norm-based (Tikhonov) techniques, ensemble-based techniques, early stopping techniques, etc. However, there is important geometric information in the learned classification function that is closely related to overfitting, and has been overlooked by previous methods. In this thesis, we study the complexity of a classification function from a new geometric perspective. In particular, we investigate the differential geometric structure in the submanifold corresponding to the estimator of the class probability P(y|x), based on the observation that overfitting produces rapid local oscillations and hence large mean curvature of this submanifold. We also show that our geometric perspective of supervised learning is naturally related to an elastic model in physics, where our complexity measure is a high dimensional extension of the surface energy in physics. This study leads to a new geometric regularization approach for supervised learning of classifiers. In our approach, the learning process can be viewed as a submanifold fitting problem that is solved by a mean curvature flow method. In particular, our approach finds the submanifold by iteratively fitting the training examples in a curvature or volume decreasing manner. Our technique is unified for both binary and multiclass classification, and can be applied to regularize any classification function that satisfies two requirements: firstly, an estimator of the class probability can be obtained; secondly, first and second derivatives of the class probability estimator can be calculated. For applications, where we apply our regularization technique to standard loss functions for classification, our RBF-based implementation compares favorably to widely used regularization methods for both binary and multiclass classification. We also design a specific algorithm to incorporate our regularization technique into the standard forward-backward training of deep neural networks. For theoretical analysis, we establish Bayes consistency for a specific loss function under some mild initialization assumptions. We also discuss the extension of our approach to situations where the input space is a submanifold, rather than a Euclidean space. / 2018-11-30T00:00:00Z